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Coding Theorems for Repetition and Superposition Codes over Binary-Input Output-Symmetric Channels

Yixin Wang, Xiao Ma

TL;DR

This work develops Repetition and Superposition (RaS) codes, a structured LDGM family, and proves that they achieve FER-capacity on binary-input output-symmetric (BIOS) channels under a new framework that also extends to source coding and JSCC. It introduces Conv-RaS as a convolutional extension with improved iterative decoding, and shows that the associated LDPC/QC-LDPC ensembles are capacity-achieving; the framework supports universal JSCC across channel, source, and joint settings. Theoretical results rely on a partial-error-exponent approach and typicality arguments, complemented by simulations indicating that Conv-RaS can operate effectively across a wide range of rates, including sub- and super-unit-rate regimes. The findings offer a flexible, universal coding paradigm that bridges channel coding, source coding, and JSCC with practical implementations and performance close to fundamental limits.

Abstract

This paper is concerned with a class of low density generator matrix codes (LDGM), called repetition and superposition (RaS) codes, which have been proved to be capacity-achieving over binary-input output-symmetric (BIOS) channels in terms of bit-error rate (BER). We prove with a recently proposed framework that the RaS codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). With this new framework, the theorem for the RaS codes can be generalized to source coding and joint source and channel coding (JSCC). In particular, we prove with this framework that the corresponding low-density parity-check (LDPC) codes, as an enlarged ensemble of quasi-cyclic LDPC (QC-LDPC) codes, can also achieve the capacity. To further improve the iterative decoding performance, we consider the convolutional RaS (Conv-RaS) code ensemble and prove it to be capacity-achieving over BIOS channels in terms of the first error event probability. The construction of Conv-RaS codes is flexible with rate (defined as the ratio of the input length to the encoding output length) ranging from less than one (typically for channel codes) to greater than one (typically for source codes), which can be implemented as a universal JSCC scheme, as confirmed by simulations.

Coding Theorems for Repetition and Superposition Codes over Binary-Input Output-Symmetric Channels

TL;DR

This work develops Repetition and Superposition (RaS) codes, a structured LDGM family, and proves that they achieve FER-capacity on binary-input output-symmetric (BIOS) channels under a new framework that also extends to source coding and JSCC. It introduces Conv-RaS as a convolutional extension with improved iterative decoding, and shows that the associated LDPC/QC-LDPC ensembles are capacity-achieving; the framework supports universal JSCC across channel, source, and joint settings. Theoretical results rely on a partial-error-exponent approach and typicality arguments, complemented by simulations indicating that Conv-RaS can operate effectively across a wide range of rates, including sub- and super-unit-rate regimes. The findings offer a flexible, universal coding paradigm that bridges channel coding, source coding, and JSCC with practical implementations and performance close to fundamental limits.

Abstract

This paper is concerned with a class of low density generator matrix codes (LDGM), called repetition and superposition (RaS) codes, which have been proved to be capacity-achieving over binary-input output-symmetric (BIOS) channels in terms of bit-error rate (BER). We prove with a recently proposed framework that the RaS codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). With this new framework, the theorem for the RaS codes can be generalized to source coding and joint source and channel coding (JSCC). In particular, we prove with this framework that the corresponding low-density parity-check (LDPC) codes, as an enlarged ensemble of quasi-cyclic LDPC (QC-LDPC) codes, can also achieve the capacity. To further improve the iterative decoding performance, we consider the convolutional RaS (Conv-RaS) code ensemble and prove it to be capacity-achieving over BIOS channels in terms of the first error event probability. The construction of Conv-RaS codes is flexible with rate (defined as the ratio of the input length to the encoding output length) ranging from less than one (typically for channel codes) to greater than one (typically for source codes), which can be implemented as a universal JSCC scheme, as confirmed by simulations.
Paper Structure (17 sections, 10 theorems, 70 equations, 8 figures)

This paper contains 17 sections, 10 theorems, 70 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation and System Model
  3. Motivation
  4. System Model
  5. Repetition and Superposition (RaS) code ensemble
  6. Main Results
  7. Coding Theorem for Systematic RaS Code Ensemble
  8. Coding Theorem for Convolutional RaS (Conv-RaS) Code Ensemble
  9. Flexibility and Universality of Conv-RaS Codes
  10. Proofs of the Main Results
  11. Preliminary Results
  12. Partial Error Exponent
  13. Proof of Coding Theorem for RaS Code Ensemble
  14. Proof of Coding Theorem for Conv-RaS Code Ensemble
  15. Simulation Results
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the RaS code ensemble defined by the generator matrices of the form $[\mathbf{I}~\mathbf{G}]$. Let $k$ and $n$ be two positive integers such that $k/(n-k)<I(X;Y)/H(U|V)$. For an arbitrarily small positive number $\epsilon$, one can always find a sufficiently large integer $m_0$ such that, f

Figures (8)

  • Figure 1: A system model with systematic linear coding.
  • Figure 2: The encoding of the RaS codes, where $\Pi_{i,j}(0\leq i\leq m, 1\leq j\leq m)$ is a random permutation matrix.
  • Figure 3: A Conv-RaS code is viewed as an asynchronous multiple access system with a tunable delay, corresponding to a flexible code rate.
  • Figure 4: The partial mutual information $I_0(p), I_1(p)$ and the mutual information $I(p)$ with $0\leq p\leq 1$ over BI-AWGN channel with BPSK with SNR $=0$ dB.
  • Figure 5: The BER performance of Conv-RaS codes for source coding. Here $k \in \{1024,2048\}$, $n-k \in \{512,640, 768, 1792\}$ and $m\in\{8, 15,20\}$. The rate is defined as $R = (n-k)/k$ and the corresponding Shannon limit is defined as the maximum $\theta \leq 1/2$ such that $H(\theta)\leq R$.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 2
  • Lemma 1
  • Example 1
  • Lemma 2
  • Lemma 3
  • ...and 1 more